Why Sensitivity-Guided Pruning Requires Two Kinds of Evidence
A calibrated QSP model can contain parameters that are poorly influential for the outputs of interest and states that appear negligible during a simulated trajectory. Either may look like a candidate for pruning. But these are different decisions. Parameter screening asks whether fixing a parameter at a specified value has a tolerable effect on selected outputs. State pruning changes the model's dynamic structure: a state may mediate a flux, feedback, or delayed response even when its concentration is small.
Sensitivity is therefore evidence for proposing a reduction, not proof that the reduced model is adequate. The assessment must specify which outputs matter, over what time window, and what error is acceptable. It must also check that the proposed equations remain mechanistically coherent, including any relevant conservation relationships, signs, and units.
Use Output-Relevant Sensitivity to Screen Parameters
For a differentiable model, the sensitivity of output \(y_k(t)\) to parameter \(\theta_j\) is \( \partial y_k(t)/\partial \theta_j \), evaluated at a stated parameter vector and input. Its units are output units per parameter unit. A large derivative may be unimportant if the parameter is only known within a narrow range; a small derivative may matter if uncertainty in the parameter is large. A useful local screening quantity therefore combines sensitivity with a plausible parameter change.
When the parameter and output are nonzero, the dimensionless relative sensitivity is \( (\theta_j/y_k)\,\partial y_k/\partial\theta_j \). This form is unsuitable when \(y_k\) is near zero. In that case, scale the output by a meaningful reference value, such as a clinically relevant threshold or a prespecified measurement scale. For time courses, inspect the sensitivity over the time window rather than relying on one selected time point. For multiple outputs, assess each key output using its own scale and tolerance.
The derivative is local: it describes a small change near the evaluation point. Earlier material on global sensitivity and alternative parameter distributions is useful when parameters vary substantially over ranges or when interactions may matter. For pruning, local results can help rank candidates, but a proposed fixed value should also be tested by directly comparing the full and reduced models over the parameter range and outcomes relevant to the analysis.
Worked Example: Screening a Weakly Influential Parameter
Hypothetical model. A biomarker output, in concentration units, is \(y=100+2\theta\), where \(\theta\) is dimensionless. At the nominal value \(\theta=1\), the output is \(102\). Suppose the plausible range is \(0.5\) to \(1.5\), and the allowed deviation from the nominal output is 2% of 102.
The local derivative is \( \partial y/\partial\theta=2 \) concentration units per unit of \(\theta\). At the nominal point, the relative sensitivity is \( (1/102)\times2=0.01961 \), or 1.961% output change per 100% parameter change. This indicates limited influence, but the finite range should be checked directly.
At the range endpoints, \(y(0.5)=100+2(0.5)=101\) and \(y(1.5)=100+2(1.5)=103\). Relative to the nominal output, the largest deviation is \( |103-102|/102=0.009804 \), or 0.9804%. The specified tolerance is \(0.02\times102=2.04\) concentration units, and the largest endpoint deviation is 1 concentration unit. Thus fixing \(\theta=1\) is a plausible screening decision for this output and range. It is not a claim that \(\theta\) is unimportant for every output or application.
Why State Pruning Needs Structural Checks
A state can affect an output indirectly through other states, even if it is not itself measured. Its importance depends on its role in the full system, not just on its trajectory magnitude. Check whether the state carries a substantial flux, controls a feedback loop, represents a conserved pool, or affects an event or threshold outcome. Also check whether removing it changes the meaning or units of any remaining term. Simply deleting a state and its equation may violate mass balance or leave an invalid expression in another equation.
A useful local justification starts from the state sensitivity equations. For fixed input and parameters, let \(S_j(t)=\partial x(t)/\partial\theta_j\). Differentiating the model with respect to \(\theta_j\) gives the linear system below, with initial condition determined by whether the initial state depends on that parameter.
This derivation shows why inspecting only the direct term \( \partial g/\partial\theta_j \) is insufficient: a parameter can change a state, which then changes the output. Similarly, a candidate state can influence outputs through downstream equations and feedback. If a state is removed, the resulting trajectory difference can be treated locally as a perturbation. For a small difference \(\delta x\), linearization gives \( \delta\dot{x}\approx A(t)\delta x+r(t) \), where \(A=\partial f/\partial x\) along the reference trajectory and \(r(t)\) represents the changed or omitted dynamics. The output difference is approximately \( \delta y\approx(\partial g/\partial x)\delta x \), plus any direct change to the output equation. A small concentration alone does not imply that \(r(t)\), its accumulated effect, or \(\delta y\) is small.
Worked Example: A Low-Concentration State With a Large Flux
Hypothetical model. A rapidly turning-over intermediate \(Z\), in amount units, is produced at 100 amount units per time and converted to mediator \(M\) at rate \(1000Z\): \( \dot{Z}=100-1000Z \), \( \dot{M}=1000Z \), with \(Z(0)=M(0)=0\). Consider the mediator amount at \(T=0.01\) time units as the key output.
Solving the first equation gives \( Z(t)=0.1(1-e^{-1000t}) \). Therefore, \(M(T)=\int_0^T1000Z(t)\,dt=100[T-(1-e^{-1000T})/1000]\). Substituting \(T=0.01\) gives \(M(T)=100[0.01-(1-e^{-10})/1000]=0.9000\) amount units (to four significant figures). At \(T\), \(Z(T)=0.1(1-e^{-10})=0.1000\) amount units.
The intermediate concentration is small in absolute terms, but its conversion flux has produced 0.9000 amount units of mediator. If pruning \(Z\) also removes its production and conversion, the reduced model has no such contribution to \(M\); the output difference is 0.9000 amount units. The state should not be removed on the basis of its concentration alone. A quasi-steady approximation or another mechanistically justified replacement might preserve its effect, but that is a different reduction proposal and must be evaluated as such.
Propose, Implement, and Test a Reduction
For each candidate, compare the full model with a clearly specified reduced model. Use the same initial conditions for retained states, inputs, parameter values, and numerical accuracy where applicable. Define an output error in units and relative to a prespecified scale. For example, for output \(k\), one may compute the maximum absolute trajectory difference \(E_k=\max_{0\le t\le T}|y_{k,\mathrm{red}}(t)-y_{k,\mathrm{full}}(t)|\), then compare it with an allowable tolerance \(\tau_k\). If the output can approach zero, an absolute tolerance is essential; a relative error alone can become unstable.
The comparison should include all outputs that justify retaining the model, including time-course features and any threshold or event quantities used in the analysis. Numerical solver tolerances should be appreciably tighter than the reduction tolerance, so numerical error is not mistaken for reduction error. Also inspect trajectories for nonphysical values, such as negative concentrations where the model requires nonnegative amounts, and check conservation relations that should hold in the reduced formulation.
Worked Example: Testing Removal of a Weakly Coupled State
Hypothetical model. Let an unobserved state \(Z\) decay as \( \dot{Z}=-Z \), with \(Z(0)=1\). A reported concentration \(Y\), initially 100 concentration units, receives a weak contribution from \(Z\): \( \dot{Y}=0.01Z \). The proposed reduced model deletes \(Z\) and its contribution, so \(Y_{\mathrm{red}}(t)=100\). Assess the output at \(T=2\) time units using an absolute tolerance of 0.02 concentration units.
In the full model, \(Z(t)=e^{-t}\), so \(Y_{\mathrm{full}}(2)=100+0.01\int_0^2e^{-t}\,dt=100+0.01(1-e^{-2})=100.00865\) concentration units. The reduced output is \(100\), giving an absolute error of \(0.008647\) concentration units. Since \(0.008647<0.02\), the proposal meets the stated tolerance for this output at this horizon.
This result supports only a limited conclusion: the removed contribution is small for the specified output and simulation. The state could still matter to another output, influence later dynamics in a different model structure, or be needed to represent a biological mechanism. Those questions require their own stated outputs and tests before pruning is adopted.
Interpret the Evidence and Choose the Right Alternative
Parameter screening and state pruning answer different questions. A parameter can often be fixed while retaining the process and its state; this reduces the number of estimated or varied quantities. State pruning changes the dynamic system and may require changing equations, outputs, and conservation relationships. A low parameter sensitivity does not establish that a connected state can be deleted, and a low state abundance does not establish that its parameters can be fixed.
| Decision | Primary evidence | Typical risk |
|---|---|---|
| Fix a parameter | Output sensitivity over a stated parameter range, followed by a direct reduced-versus-full comparison | Local insensitivity or parameter interactions hide effects elsewhere |
| Remove a state | Mechanistic role, flux and feedback checks, and output comparison after specifying replacement equations | Deleting an indirect pathway, breaking conservation, or losing a transient response |
| Eliminate or aggregate a state | An exact relationship or a justified reduced representation | Assuming an approximation remains valid outside its intended conditions |
Exact algebraic elimination, state aggregation, and trajectory-based reduction can be preferable when simple deletion would discard a meaningful process. Each preserves different information and has different conditions. Sensitivity-guided pruning is most useful as a disciplined way to identify candidates and quantify consequences, not as a substitute for those structural methods.
Diagnostics, Limitations, and a Common Mistake
- Check the output definition. A parameter or state can be negligible for one endpoint and influential for a time-course feature or event outcome.
- Check the sensitivity scale. A derivative in raw units can be misleading across parameters or outputs with different units. Use plausible parameter changes and output-specific scales.
- Check range and interactions. Local derivatives can miss curvature, threshold behavior, and interactions. Directly evaluate the proposed fixed value and relevant range; use broader sensitivity analysis when appropriate.
- Check mechanics and numerics. Inspect fluxes, feedback, balances, positivity, solver accuracy, and whether the reduced equations remain well-defined.
Common mistake: “State \(Z\) stays near zero, so deleting it cannot matter.” This reasoning confuses state magnitude with influence. The correction is to inspect how \(Z\) enters fluxes and feedback, then compare the full and reduced outputs under an explicit tolerance. A small pool can turn over rapidly and carry a large cumulative flux.
Sensitivity-guided pruning is limited by the chosen outputs, parameter ranges, and conditions. Passing a reduction test for one simulated setting does not establish adequacy for every use of the model. Record what was pruned, what was retained, the reference values or replacement equations, the outputs assessed, and the tolerances used. This makes the reduction decision interpretable and reproducible.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A parameter has a small local derivative at its nominal value. Why is that not, by itself, enough to justify fixing it?
- A state concentration is low, but its conversion rate is proportional to a very large rate constant. What should be checked before pruning it?
- For a key output, the full-model value is 20 units and the reduced-model value is 20.3 units. If the absolute tolerance is 0.5 units, does this comparison pass?
- Why can an output-relative error be unsuitable when the full-model output is near zero?
- State one distinction between fixing a parameter and pruning a state.
Solutions. (1) The derivative is local; plausible parameter variation, curvature, interactions, and other key outputs may change the conclusion. Directly compare the proposed fixed-value model over the relevant range. (2) Inspect the flux, cumulative downstream effect, feedback, and output differences after specifying the reduced equations; small concentration alone is not evidence of negligible influence. (3) Yes, for this comparison: the absolute difference is \(|20.3-20|=0.3\) units, and \(0.3<0.5\). (4) Division by a value near zero can produce an unstable or misleading relative error, so use an appropriate absolute scale or tolerance. (5) Fixing a parameter reduces parameter variation or estimation while retaining the dynamic state; pruning a state changes the model's dynamic structure and may require revised equations.